nLab
cosmos

Idea

A cosmos is a “good place in which to do category theory,” including both ordinary and enriched category theory.

The word is chosen by analogy with topos which can be regarded as “a good place to do set theory,” but there are notable differences between the two situations; a more direct categorification of a topos is, unsurprisingly, a 2-topos?. In contrast, cosmoi also include enriched category theory, while toposes do not allow non-cartesian enrichment.

There are a number of different, inequivalent, definitions of “cosmos” in the literature.

Benabou’s definition

Jean Bénabou's original definition was that a cosmos $V$ is a complete and cocomplete closed symmetric monoidal category. This is an ideal situation for studying categories enriched over $V$.

Street’s “fibrational cosmoi”

Ross Street has taken a different tack, defining a “cosmos” to be the collection of (enriched) categories and relevant structure for doing category theory, rather than the “base” category $V$ over which the enrichment occurs.

In his paper “Elementary cosmoi,” Street defined a (fibrational) cosmos to be a 2-category in which internal fibrations are well-behaved and representable by a structure of “presheaf objects” (later realized to be a special sort of Yoneda structure?). Note that while this includes $\mathrm{Cat}$, it does not include $V$-$\mathrm{Cat}$ for non-cartesian $V$, since internal fibrations are poorly behaved there.

Street’s second definition

In his paper “Cauchy characterization of enriched categories,” Street instead defined a “cosmos” to be a 2-category that behaves like the 2-category $V$-$\mathrm{Mod}$ of enriched categories and profunctors. The precise definition: a cosmos is a 2-category (or bicategory) such that:

• Small (weak, or bi-) coproducts exist.
• Each monad admits a Kleisli construction? (analogous to the exactness of a topos).
• It is locally small-cocomplete, i.e. its hom-categories have small colimits that are preserved by composition on each side.
• There exists a small “Cauchy generator”.

These hypotheses imply that it is equivalent to the bicategory of categories and profunctors enriched over some “base” bicategory. (Note the generalization from enrichment over a monoidal category to enrichment over a bicategory.)

Defined in this way, cosmoi are closed under dualization, parametrization and localization, suitably defined.